AAE 3710
Fundamentals of Aerodynamics
Lecture 2 Static Fluid
01/18/2006
Some math preliminaries
Continuous function expressed in power series
For engineering analysis,
these higher-than-first-order
term can be neglected resulting
in 2nd order approximation to f(x)
about point a
n −1
(
( x − a)3
x−a
x − a)
( x − a)
''
'''
f ( x) = f (a ) + f (a )
+ f (a)
+ f (a)
+ ... + f n −1 (a )
+ H .O.T .
1!
2!
3!
(n − a )!
Some expansions of typical functions:
2
'
1 1 1
e = 1 + + + + ...
1! 2! 3!
x3 x5 x7
sin x = x − + − + ...
3! 5! 7!
Vectorial differential symbols
Differential operator
∇ =
∂ r
∂
∂ r
i +
j+
k
∂x
∂y
∂z
Gradient of a scalar function
∇f =
∂f r ∂f
∂f r
i+
j+ k
∂x
∂y
∂z
normal to the surface
f=c; the direction of
the greatest change
in function f; the
magnitude of the
gradient is the
magnitude of this
change.
Some math preliminaries
Differential analysis
In engineering, the analysis of a complex phenomena can often be
easily carried out by examining a very small/differential chunk taken
out of the original object.
Reason: flexible approximation schemes can be applied to the very
small sample.
Static Fluid
Stationary fluid = at rest or no relative motion between adjacent
particles
No shearing forces in the fluid;
behave as a single object.
A bottle of coke sitting on a table;
Bottles of coke on smoothly moving
conveyor belt.
Pressure is responsible for the surface forces
Behavior of Pressure in Static Fluid
Pressure at a point
In absence of shearing force
Py = Pz = Ps
The pressure at a point in a
fluid at rest, or in motion, is
independent of direction as
long as there are no shearing
stresses present
Behavior of Pressure in Static Fluid
Pressure distribution in a fluid field
Using first 2 terms in Taylor series expansion,
the pressure function P(x, y, z) can be extrapolated
onto its 6 neighboring planes
The resultant surface forces (or traction)
In there dimensional directions.
differential operator
− ∇Pδxδyδz
Behavior of Pressure in Static
Fluid
Pressure distribution in a fluid field
Body force for this case is only due to
the weight of the element
Newton’s 2nd law:
substitution
Eqn. of Motion for
A liquid without
shearing stresses
Fluid at
rest
Fluid in
rigid-body motion
Behavior of Pressure in Static Fluid
Fluid at rest
Velocity is zero
a=0
Velocity is constant with time
i
In a fluid without acceleration:
Pressure does not change with x nor y;
Pressure is constant on the plane perpendicular
to z axis;
For different planes perpendicular to z axis, the pressure
will be different, the rate of change is the negative specific weight.
pr
m
o
nc
le
ib
s
es
Behavior of Pressure in Static Fluid
Pressure head
h represents the height of a
column of fluid of specific
weight required to give a
pressure difference P1-P2
Example:
23.1 ft water head tells you the pressure difference is 10 psi
Pressure in the container of arbitrary shape
What is the pressure on plane A-B?
Ans:
No ge
omet
ric
in this
varia
bles
.
eqn
h means the vertical
distance between two
surfaces
Behavior of Pressure in Static Fluid
Comprssible fluid
From eqn. of motion
Ideal gas law
If T is constant
nger
is no lo e
n
io
t
th
ria
ted to
ure va
Press linearly rela hen the
w
simply difference
n
io
le
t
a
essib
Elev
compr
fluid is
Standard Atmosphere
The earth’s atmosphere is a dynamically
changing system, many influential factors:
altitude, longitude, latitude, time of day,
season, solar sunspot activity…
Purpose
of
the definition of Standard atmosphere
T=T1+a(h-h1)
To relate flight tests, wind tunnel results,
and general airplane design and
performance to a common reference
Thru experiments,
Balloons, sounding
Roket measurements
Mean values of pressure, temperature,
density, and other properties as functions
of altitude
ρ ⎛T ⎞
=⎜ ⎟
ρ1 ⎜⎝ T1 ⎟⎠
−{g 0 /( aR ) +1}
Measurement of Pressure
Read section 2.5, 2.6 and 2.7.
P(abs)
P(abs) + rho(H2O)*SG(oil)*g*(h1+h2) = P(atm) +rho(H2O)*SG(Hg)*g*h3
P(abs) – P(atm) =- rho(H2O)*SG(oil)*g*(h1+h2) + rho(H2O)*SG(Hg)*g*h3
P(gage) = P(abs) – P(atm)
It is important to
determine a reference
surface.
A Little Road Map
Static Fluids
Eqn. of Motion
Fluid at rest
Exponential
relation
Linear relation
Incompressible
flow
Pressure
At a point
Fluid in rigid body motion
Compressible flow
Pressure
On a surface
?
Hydrostatic Force on a Plane Surface
Pressure varies linearly with the latitude difference in an incompressible fluids
Centroid
measured
from x axis
1. Magnitude of the resultant force is
the pressure at the centroid of the area
multiplied by the total area
2. Magnitude of the resultant force is only
dependent on specific weight of the fluid;
fluid
total area;
area depth of the centroid of the area
below the surface
Hydrostatic Force on a Plane Surface
To determine the location of the
resultant force FR, the equlibrium
of the moment is enforced
<
Hydrostatic Force on a Plane Surface
Geometric properties of some common shapes
Hydrostatic Force on a Plane Surface
Magn
it
Resu ude of
ltant f
orce is
The p
re
Time ssure at t
s the
he ce
area
ntroid
Hydrostatic Force on a Plane Surface